Proof. We shall assume that in each of the three cases \(n = 15\) , \(n = 16\)  and \(n \ge 17\)  we consider, \(G\)  as a graph on \(5(n-1)\)  vertices satisfying \(C_n \lnot \subseteq G\)  and \(\alpha (G) \le 5\) . Since \(r(C_{n-1}, K_6)=5n-9 \le 5(n-1)\)  (see [1]}, [2]}), there exists a cycle \(C=(u_1, u_2, ... , u_{n-1}, u_1)\)  of length \(n - 1\)  in \(G\) . In consistent with the notation of [1]}, define \(H\)  as the induced subgraph of \(G\)  not containing the vertices of the cycle \(C\) . Then, \(|V(C)|=n-1\)  and \(|V(H)|=4(n-1)\) .
